X2093

REFLECTION OF X1 IN X57

X2093 is the reflection of X1 in X57.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 1, 57 }: OI Line (Incenter–Circumcenter Line) { 1, 3 }: OI Line (Incenter–Circumcenter Line) { 10, 329 }: Central Line L(10, 329) { 200, 758 }: Central Line L(200, 758) { 553, 1056 }: Central Line L(553, 1056) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_329 = ISOTOMIC CONJUGATE OF X(189) (0.3847, -0.0222) X_200 = -CEVA CONJUGATE OF X(9) (0.4392, 0.1037) X_758 = EVEN (1, 3) INFINITY POINT (0.1821, -0.2671) X_553 = POINT MAIA II (0.5963, 0.2857) X_1056 = POINT ALCYONE (0.5756, 0.2499) X_2093 = REFLECTION OF X(1) IN X(57) (0.6561, 0.3888) A B C X1 X57 X3 X10 X329 X200 X758 X553 X1056 X2093 (a, b, c) = (5, 6, 7) | AB = 1

X2093 (REFLECTION OF X1 IN X57) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(a^{3} + 3(b + c)a^{2} - (b + c)^{2}a - 3(b + c)(b - c)^{2}\)
Search Value (6, 9, 13): \(-0.929397926422\)
▸ Alternate Coordinates1 form

  • Trilinears: \(\cos A - 3 \cos B - 3 \cos C + 1 : :\)

▸ Incident Central Lines16 lines

{1, 3}, {10, 329}, {19, 1743}, {72, 1706}, {196, 1785}, {200, 758}, {515, 2096}, {518, 2097}, {519, 2094}, {527, 1478}, {553, 1056}, {946, 1788}, {962, 1210}, {1046, 1719}, {1103, 1254}, {1699, 1737}

▸ Functional Relations2 relations

Notes

Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is homothetic to the 2nd isogonal triangle of X1 at X2093. (Randy Hutson, December 2, 2017)